Weak sequential completeness and related properties of some operator spaces.

*(Russian)*Zbl 0565.47029Three problems are investigated for Banach ideals of p-nuclear and p- quasi-nuclear operators. [Y. Gordon, D. R. Lewis and J. R. Retherford, J. Funct. Anal. 14, 85-129 (1973; Zbl 0272.47024).] The first is weak completeness for \(1<p<\infty\). The authors improve previous results by removing the assumption of an unconditional basis on one of the spaces \(X^*\), Y. In the case of p-quasi-nuclear operators, \(QN_ p(X,Y)\), they also remove the assumption of an approximation property.

The main result is: Let \(X^*\), Y be weakly complete, \(1<p<\infty\), and \(QN_ p(X,Y)=\Pi_ p(X,Y)\) (the ideal of p-absolutely summing operators). Then the space \(QN_ p(X,Y)\) is weakly complete. They also observe that if Y contains an unconditional basis, the assumption \(QN_ p=\Pi_ p\) is necessary. Similar results are obtained for the p-nuclear operators with the assumptions that \(N_ p(X,Y)=I_ p(X,Y)\) (the ideal of p-integral operators) and that \(Y^*\) has the approximation property.

Secondly, they present sufficient conditions under which \(N_ p(X,Y)\) does not contain \(\ell_ 1\). For \(1<p<\infty\) they show that this holds if \(X^*\) does not contain \(\ell_ 1\), \(Y^*\) has the Radon-Nikrondym property, and one of \(X^{**}\), \(Y^{**}\) has the approximation property. For \(p=1\) they require Y reflexive and X not containing \(\ell_ 1\). They also obtain corollaries about p-tensor products as defined by (INVALID INPUT)P. Saphar [Stud. Math. 38, 71-100 (1970; Zbl 0213.142)]. Lastly, they obtain sufficient conditions under which these ideals do not contain \(c_ 0\). For example, with \(1\leq p<\infty\), if \(X^*\), Y do not contain \(c_ 0\) and \(QN_ p(X,Y)=\Pi_ p(X,Y)\), then \(QN_ p(X,Y)\) does not contain \(c_ 0\).

The main result is: Let \(X^*\), Y be weakly complete, \(1<p<\infty\), and \(QN_ p(X,Y)=\Pi_ p(X,Y)\) (the ideal of p-absolutely summing operators). Then the space \(QN_ p(X,Y)\) is weakly complete. They also observe that if Y contains an unconditional basis, the assumption \(QN_ p=\Pi_ p\) is necessary. Similar results are obtained for the p-nuclear operators with the assumptions that \(N_ p(X,Y)=I_ p(X,Y)\) (the ideal of p-integral operators) and that \(Y^*\) has the approximation property.

Secondly, they present sufficient conditions under which \(N_ p(X,Y)\) does not contain \(\ell_ 1\). For \(1<p<\infty\) they show that this holds if \(X^*\) does not contain \(\ell_ 1\), \(Y^*\) has the Radon-Nikrondym property, and one of \(X^{**}\), \(Y^{**}\) has the approximation property. For \(p=1\) they require Y reflexive and X not containing \(\ell_ 1\). They also obtain corollaries about p-tensor products as defined by (INVALID INPUT)P. Saphar [Stud. Math. 38, 71-100 (1970; Zbl 0213.142)]. Lastly, they obtain sufficient conditions under which these ideals do not contain \(c_ 0\). For example, with \(1\leq p<\infty\), if \(X^*\), Y do not contain \(c_ 0\) and \(QN_ p(X,Y)=\Pi_ p(X,Y)\), then \(QN_ p(X,Y)\) does not contain \(c_ 0\).

Reviewer: A.J.Klein

##### MSC:

47L10 | Algebras of operators on Banach spaces and other topological linear spaces |

47L05 | Linear spaces of operators |

46B20 | Geometry and structure of normed linear spaces |